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R0019-10016 · q-20261003-r03-factorization-0091
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(15)² equal to the proposed coefficient vector [-224, 0, 1]? Answer Yes or No.
G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(15)² equal to the proposed coefficient vector [-224, 0, 1]? Answer Yes or No.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
Q[x];ascending coefficients;input vectors may have trailing zeros
After
equality uses every normalized coefficient,not numerical sampling
Conditions: none
Preserve the original factor order and vectors.Normalization removes only trailing high-power zeros;do not reorder source factors by semantic identity or infer polynomial equality from one point.
Rule: factorization_coefficient_identity_scope · Uses: G001
Before
original d=15;x^2-d^2
After
factors x-d and x+d;ascending vectors=[[-15, 1], [15, 1]]
Conditions: none
Expand(x−d)(x+d)=x²+dx−dx−d²,retaining the original sign of d.The same expansion applies for negative d or d=0.
Rule: difference_of_squares_actual_signed_parameter · Uses: G001, S001
Before
power k=0;left=[-15, 1];right=[15, 1]
After
all index pairs i+j=k=[[0, 0]]
Conditions: none
Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication.
Rule: product_coefficient_complete_index_pairs · Uses: G001, S002
Before
coefficient0 term0,0 := mul(-15, 15)
After
coefficient0 term0,0 = -225
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
coefficient0 complete sum := sum(-225)
After
coefficient0 complete sum = -225
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
power k=1;left=[-15, 1];right=[15, 1]
After
all index pairs i+j=k=[[0, 1], [1, 0]]
Conditions: none
Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication.
Rule: product_coefficient_complete_index_pairs · Uses: G001, S005
Before
coefficient1 term0,1 := mul(-15, 1)
After
coefficient1 term0,1 = -15
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
coefficient1 term1,0 := mul(1, 15)
After
coefficient1 term1,0 = 15
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
coefficient1 complete sum := sum(-15, 15)
After
coefficient1 complete sum = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
power k=2;left=[-15, 1];right=[15, 1]
After
all index pairs i+j=k=[[1, 1]]
Conditions: none
Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication.
Rule: product_coefficient_complete_index_pairs · Uses: G001, S009
Before
coefficient2 term1,1 := mul(1, 1)
After
coefficient2 term1,1 = 1
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
coefficient2 complete sum := sum(1)
After
coefficient2 complete sum = 1
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
all collected product coefficients=[-225, 0, 1]
After
normalized complete product=[-225, 0, 1]
Conditions: none
All finitely many coefficients were obtained.Remove trailing high-power zeros and display zero as[0].This is coefficient identity,not agreement at a few sample points.
Rule: complete_product_not_sampled_values · Uses: G001, S012
Before
power k=0;actual coefficient=-225;candidate coefficient=-224
After
coefficient matches=No
Conditions: none
Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product.
Rule: every_identity_coefficient_comparison · Uses: G001, S013
Before
power k=1;actual coefficient=0;candidate coefficient=0
After
coefficient matches=Yes
Conditions: none
Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product.
Rule: every_identity_coefficient_comparison · Uses: G001, S014
Before
power k=2;actual coefficient=1;candidate coefficient=1
After
coefficient matches=Yes
Conditions: none
Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product.
Rule: every_identity_coefficient_comparison · Uses: G001, S015
Before
raw proposed=[-224, 0, 1];normalized proposed=[-224, 0, 1];actual product=[-225, 0, 1]
After
coefficient identity=No
Conditions: none
Identity requires equality of all coefficients,including the both-zero case.Constants and trailing-zero representations do not create a different polynomial.
Rule: factorization_final_full_identity_gate · Uses: G001, S016
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute coefficient2 complete sum with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute coefficient2 term1,1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute coefficient1 complete sum with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute coefficient1 term1,0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute coefficient1 term0,1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute coefficient0 complete sum with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute coefficient0 term0,0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.No
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c039bd496ef8ad120c56baf802b884a9d0b9c7b56b8846f2fede084b9edca993candidate record SHA-256: 0ea687ca58369bc73d672df6b2e330c02c36182c701306da118e77eece78d035semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED